Properties

Label 4.223952.5t5.a
Dimension $4$
Group $S_5$
Conductor $223952$
Indicator $1$

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:\(223952\)\(\medspace = 2^{4} \cdot 13997 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 5.5.223952.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Projective image: $S_5$
Projective field: Galois closure of 5.5.223952.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 26 + 146\cdot 173 + 116\cdot 173^{2} + 98\cdot 173^{3} + 84\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 53 + 48\cdot 173 + 14\cdot 173^{2} + 25\cdot 173^{3} + 164\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 58 + 145\cdot 173 + 138\cdot 173^{2} + 122\cdot 173^{3} + 44\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 100 + 94\cdot 173 + 35\cdot 173^{2} + 110\cdot 173^{3} + 49\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 109 + 84\cdot 173 + 40\cdot 173^{2} + 162\cdot 173^{3} + 2\cdot 173^{4} +O(173^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character values
$c1$
$1$ $1$ $()$ $4$
$10$ $2$ $(1,2)$ $2$
$15$ $2$ $(1,2)(3,4)$ $0$
$20$ $3$ $(1,2,3)$ $1$
$30$ $4$ $(1,2,3,4)$ $0$
$24$ $5$ $(1,2,3,4,5)$ $-1$
$20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.